# Simplify (4+ square root of 5)/(4- square root of 5)+(4- square root of 5)/(4+ square root of 5) 4+54-5+4-54+5
Simplify terms.
Simplify each term.
Multiply 4+54-5 by 4+54+5.
4+54-5⋅4+54+5+4-54+5
Multiply 4+54-5 and 4+54+5.
(4+5)(4+5)(4-5)(4+5)+4-54+5
Expand the denominator using the FOIL method.
(4+5)(4+5)16+45-45-52+4-54+5
Simplify.
(4+5)(4+5)11+4-54+5
Simplify the numerator.
Raise 4+5 to the power of 1.
(4+5)1(4+5)11+4-54+5
Raise 4+5 to the power of 1.
(4+5)1(4+5)111+4-54+5
Use the power rule aman=am+n to combine exponents.
(4+5)1+111+4-54+5
Add 1 and 1.
(4+5)211+4-54+5
(4+5)211+4-54+5
Rewrite (4+5)2 as (4+5)(4+5).
(4+5)(4+5)11+4-54+5
Expand (4+5)(4+5) using the FOIL Method.
Apply the distributive property.
4(4+5)+5(4+5)11+4-54+5
Apply the distributive property.
4⋅4+45+5(4+5)11+4-54+5
Apply the distributive property.
4⋅4+45+5⋅4+5511+4-54+5
4⋅4+45+5⋅4+5511+4-54+5
Simplify and combine like terms.
Simplify each term.
Multiply 4 by 4.
16+45+5⋅4+5511+4-54+5
Move 4 to the left of 5.
16+45+4⋅5+5511+4-54+5
Combine using the product rule for radicals.
16+45+45+5⋅511+4-54+5
Multiply 5 by 5.
16+45+45+2511+4-54+5
Rewrite 25 as 52.
16+45+45+5211+4-54+5
Pull terms out from under the radical, assuming positive real numbers.
16+45+45+511+4-54+5
16+45+45+511+4-54+5
Add 16 and 5.
21+45+4511+4-54+5
Add 45 and 45.
21+8511+4-54+5
21+8511+4-54+5
Multiply 4-54+5 by 4-54-5.
21+8511+4-54+5⋅4-54-5
Multiply 4-54+5 and 4-54-5.
21+8511+(4-5)(4-5)(4+5)(4-5)
Expand the denominator using the FOIL method.
21+8511+(4-5)(4-5)16-45+5⋅4-52
Simplify.
21+8511+(4-5)(4-5)11
Simplify the numerator.
Raise 4-5 to the power of 1.
21+8511+(4-5)1(4-5)11
Raise 4-5 to the power of 1.
21+8511+(4-5)1(4-5)111
Use the power rule aman=am+n to combine exponents.
21+8511+(4-5)1+111
Add 1 and 1.
21+8511+(4-5)211
21+8511+(4-5)211
Rewrite (4-5)2 as (4-5)(4-5).
21+8511+(4-5)(4-5)11
Expand (4-5)(4-5) using the FOIL Method.
Apply the distributive property.
21+8511+4(4-5)-5(4-5)11
Apply the distributive property.
21+8511+4⋅4+4(-5)-5(4-5)11
Apply the distributive property.
21+8511+4⋅4+4(-5)-5⋅4-5(-5)11
21+8511+4⋅4+4(-5)-5⋅4-5(-5)11
Simplify and combine like terms.
Simplify each term.
Multiply 4 by 4.
21+8511+16+4(-5)-5⋅4-5(-5)11
Multiply -1 by 4.
21+8511+16-45-5⋅4-5(-5)11
Multiply 4 by -1.
21+8511+16-45-45-5(-5)11
Multiply -5(-5).
Multiply -1 by -1.
21+8511+16-45-45+15511
Multiply 5 by 1.
21+8511+16-45-45+5511
Raise 5 to the power of 1.
21+8511+16-45-45+51511
Raise 5 to the power of 1.
21+8511+16-45-45+515111
Use the power rule aman=am+n to combine exponents.
21+8511+16-45-45+51+111
Add 1 and 1.
21+8511+16-45-45+5211
21+8511+16-45-45+5211
Rewrite 52 as 5.
Use axn=axn to rewrite 5 as 512.
21+8511+16-45-45+(512)211
Apply the power rule and multiply exponents, (am)n=amn.
21+8511+16-45-45+512⋅211
Combine 12 and 2.
21+8511+16-45-45+52211
Cancel the common factor of 2.
Cancel the common factor.
21+8511+16-45-45+52211
Divide 1 by 1.
21+8511+16-45-45+5111
21+8511+16-45-45+5111
Evaluate the exponent.
21+8511+16-45-45+511
21+8511+16-45-45+511
21+8511+16-45-45+511
Add 16 and 5.
21+8511+21-45-4511
Subtract 45 from -45.
21+8511+21-8511
21+8511+21-8511
21+8511+21-8511
Combine the numerators over the common denominator.
21+85+21-8511
21+85+21-8511
Simplify the numerator.
Add 21 and 21.
42+85-8511
Subtract 85 from 85.
42+011
Add 42 and 0.
4211
4211
The result can be shown in multiple forms.
Exact Form:
4211
Decimal Form:
3.81‾
Mixed Number Form:
3911
Simplify (4+ square root of 5)/(4- square root of 5)+(4- square root of 5)/(4+ square root of 5)

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